Complex varieties and higher integrability of Dir-minimizing Q-valued functions

TL;DR

This paper offers new elementary proofs showing that complex varieties can be locally represented as Dir-minimizing functions and establishes higher integrability of their gradients, with optimal exponents in the planar case.

Contribution

It provides simplified proofs of known results and demonstrates the optimality of integrability exponents in the planar setting.

Findings

Every complex variety is locally a graph of a Dir-minimizing function.

Gradients of Dir-minimizing functions are p-integrable for some p > 2.

In the planar case, the integrability exponents are proven to be optimal.

Abstract

We provide new elementary proofs of the following two results: every complex variety is locally the graphs of a Dir-minimizing function, first proved by Almgren; the gradients of Dir-minimizing functions, in principle square-summable, are p-integrable for some p > 2, proved by De Lellis and the author. In the planar case, we prove that our integrability exponents are optimal.